Answer
$\angle1,\angle9,\angle10=55^{\circ}$
$\angle2,\angle4=65^{\circ}$
$\angle3,\angle5,\angle7,\angle8=60^{\circ}$
$\angle6=120^{\circ}$
Work Step by Step
1. We will find angles in not numerical order. The easiest way is to start from finding vertical angles of labeled angles (two intersecting lines and angles opposite each other)
$\angle6=120^{\circ}$ and $\angle1=55^{\circ}$
2. Angle 8 and angle labeled 120 degrees make up a straight line, so they must add up to 180 degrees, therefore:
$\angle8=180^{\circ}-120^{\circ}=60^{\circ}$
3. Angle 7 and 8 are vertical, therefore congruent
$\angle7=\angle8=60^{\circ}$
4. Angle 10 is an alternate interior angle with the angle labeled 55 degrees
$\angle10=55^{\circ}$
5. Angles 9 and 10 are vertical angles therefore
$\angle9=\angle10=55^{\circ}$
6. Angles 5 and 8 are corresponding angles
$\angle5=\angle8=60^{\circ}$
7. Angles 3 and 5 are vertical
$\angle3=\angle5=60^{\circ}$
8. Angle 4 makes a straight line with angles 5, and labeled 55, so they must add up to 180'degrees.
$\angle4= 180-60-55=65^{\circ}$
9. Also $\angle2=\angle4=65^{\circ}$
